Related papers: On the logarithimic calculus and Sidorenko's conje…
An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined…
We study pairs of graphs (H_1,H_2) such that every graph with the densities of H_1 and H_2 close to the densities of H_1 and H_2 in a random graph is quasirandom; such pairs (H_1,H_2) are called forcing. Non-bipartite forcing pairs were…
In this paper, we study the well-known Kohayakawa-Nagle-R{\"o}dl-Schacht (KNRS) conjecture, with a specific focus on graph subdivisions. The KNRS conjecture asserts that for any graph $H$, locally dense graphs contain asymptotically at…
We prove the following version of the Loebl-Komlos-Sos Conjecture: For every alpha>0 there exists a number M such that for every k>M every n-vertex graph G with at least (0.5+alpha)n vertices of degree at least (1+alpha)k contains each tree…
We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.
We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted)…
A vertex in a directed graph is said to have a large second neighborhood if it has at least as many second out-neighbors as out-neighbors. The Second Neighborhood Conjecture, first stated by Seymour, asserts that there is a vertex having a…
Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most…
In 2004, Karo\'nski, \L uczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper. After that, the total…
We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is…
We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs.…
In \cite{reed97}, Reed conjectures that the inequality $\chi (G) \leq \left \lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From…
We provide a very elementary proof of a classical extremality result due to Sidorenko (Discrete Math. 131.1-3, 1994), which states that among all connected graphs $G$ on $k$ vertices, the $k$-vertex star maximises the number of graph…
The famous Tetrahedron Conjecture of Tur\'an from the 1940s asserts that the number of edges in an $n$-vertex $3$-graph without the tetrahedron, the complete $3$-graph on four vertices, cannot exceed that of the balanced complete cyclic…
We prove part of a conjecture by Johansson, Kahn and Vu \cite{JKV} regarding threshold functions for the existence of an $H$-factor in a random graph \gnp. We prove that the conjectured threshold function is correct for any graph $H$ which…
Many results in extremal graph theory can be formulated as certain polynomial inequalities in graph homomorphism densities. Answering fundamental questions raised by Lov{\'a}sz, Szegedy and Razborov, Hatami and Norine proved that…
The 1-2-3 Conjecture, posed by Karo\'{n}ski, {\L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The…
A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev's theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight…
Karo\'nski, {\L}uczak and Thomason conjectured in 2004 that for every finite graph without isolated edge, the edges can be assigned weights from $\{1,2,3\}$ in such a way that the endvertices of each edge have different sums of incident…
In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 -…